The simplest and most common macro-scale mechanical test is the tensile (or compression) test. In this test, a sample of uniform cross section is stretched (or compressed) uniaxially while the resulting deformation of the sample is continuously monitored. The stress a is defined as the force applied to the sample divided by the cross-sectional area of the sample, and the strain E is defined as the change in length of the sample divided by the original length. A typical plot from such a test is shown in FIG. 1.
The strain increases in proportion to the stress while the deformation remains elastic, with the constant of proportionality being the Young's Modulus of the material, E. The onset of plasticity is identified as the yield point, or the point at which the strain begins to increase disproportionately to the stress. The yield point is designated “Y” in FIG. 1. Beyond the yield point Y, a variety of micro-structural mechanisms determine the relation between stress and strain; these include dislocation motion and entanglement, grain boundary sliding, micro-fracturing, etc.
Due to the prevalence of the tensile test, many mechanical computational models require, as input, the stress-strain curve of the material.
However, the design and manufacturing of products on a small scale, e.g., where dimensions may be on the order of microns, cannot be accomplished using properties of materials measured with macro-sized samples because the properties of a material depends on its micro-structure, which in turn depends on the scale of the material. Accordingly, instrumented indentation has emerged as the premier method for measuring mechanical properties of small scale samples (i.e., small volumes of materials).
Standardized instrumented indentation can be used to determine the Young's modulus as well as the hardness of a small volume of material.
One class of indentation equipment or “nanoindenters” for carrying out instrumented indentation commonly employs a geometrically self-similar indenter punch, e.g., a self-similar pyramidal indenter punch known as a Berkovich indenter tip. A Berkovich indenter tip, however, can impose only one effective strain on the test material. Thus, a Berkovich indenter tip cannot be used to determine the relationship between stress and strain beyond the yield point.
Nanoindenters having a spherical type of indenter have been used to derive the mechanical stress-strain curve with limited success. With these indenters, a spherical surface of an indenter punch is forced into the test material, and the strain imposed on the test material increases with the indentation force. When the spherical surface of the indenter punch first contacts the surface of the test material, the strain is small, and the deformation is elastic. The imposed strain increases as the indenter punch is pressed further into the test material, and eventually causes plastic yield in the test material. David Tabor demonstrated, circa 1956, that the hardness measured with a spherical indenter could be scaled so as to overlay the true stress-strain curve for the material. However, a number of practical difficulties plague spherical indentation. Most importantly, the initial onset of plasticity is difficult to detect, because plastic yield nucleates below the surface at the point of maximum shear stress, and material which has plastically yielded is initially constrained by elastic material. Thus, the proportionality discovered by Tabor is only valid at relatively large strains. Furthermore, as the contact area grows, the volume of material being tested also grows. Thus, both the material volume and the strain are changing concurrently. This difficulty is not insurmountable if the test material is homogenous, but if the material is substantially heterogeneous, then the problem of simultaneously changing both the strain and the material being tested is quite intractable. Generally, uncertainty in contact area is also greater for spheres than for other indenter punch geometries.
Instrumented indentation equipment having a cylindrical indenter punch with a flat-ended surface, namely, a flat-ended indenter, has been proposed, and is commonly used for such purposes as measuring the viscoelastic properties of polymers and biological materials.
Moreover, instrumented indentation equipment having flat-ended and spherical indenters each have been used to perform compression tests on very small pillars fabricated by focused ion-beam milling (FIB) or other micro-fabrication techniques. In these tests, the contact surface of the indenter punch is brought into contact with the top of the “micro-pillar”, and then compresses the pillar to the yield point and beyond. The analysis of the indentation force and displacement data generated by this kind of test is identical to that of the force-displacement data produced by a macro-scale compression test, and each test on an individual micro-pillar can be used to construct a full stress-strain curve.
The growing popularity of the pillar-compression technique reveals a great deal about the value and challenge of measuring stress-strain curves on microscopic samples. Producing micro-pillars is time-consuming, and requires expensive equipment and highly experienced operators. Therefore, the fact that those skilled in the art choose to conduct instrumented indentation on micro-pillars demonstrates that no adequate alternative instrumented indentation equipment and techniques presently exist.
Thus, an economical and efficient way of determining the entire stress-strain curve of a sample of material on a scale applicable to nanotechnology by means of instrumented indentation remains a highly desirable but elusive goal.